Geometric relative entropies and barycentric Rényi divergences

Abstract

We give systematic ways of defining monotone quantum relative entropies and (multi-variate) quantum Rényi divergences starting from a set of monotone quantum relative entropies. Despite its central importance in information theory, only two additive and monotone quantum extensions of the classical relative entropy have been known so far, the Umegaki and the Belavkin-Staszewski relative entropies. Here we give a general procedure to construct monotone and additive quantum relative entropies from a given one with the same properties; in particular, when starting from the Umegaki relative entropy, this gives a new one-parameter family of monotone and additive quantum relative entropies interpolating between the Umegaki and the Belavkin-Staszewski ones on full-rank states. In a different direction, we use a generalization of a classical variational formula to define multi-variate quantum Rényi quantities corresponding to any finite set of quantum relative entropies $(D^{q_x})_{x\in X}$ and signed probability measure $P$, as $$ Q_P^{\mathrm{b},\mathbf{q}}((ρ_x)_{x\in X}):=\sup_{τ\ge 0}\left\{\text{Tr}\,τ-\sum_xP(x)D^{q_x}(τ\|ρ_x)\right\}. $$ We show that monotone quantum relative entropies define monotone Rényi quantities whenever $P$ is a probability measure. With the proper normalization, the negative logarithm of the above quantity gives a quantum extension of the classical Rényi $α$-divergence in the 2-variable case ($X=\{0,1\}$, $P(0)=α$). We show that if both $D^{q_0}$ and $D^{q_1}$ are monotone and additive quantum relative entropies, and at least one of them is strictly larger than the Umegaki relative entropy then the resulting barycentric Rényi divergences are strictly between the log-Euclidean and the maximal Rényi divergences, and hence they are different from any previously studied quantum Rényi divergence.